紧集与相对分形鼓的分形管公式：振荡、复维数与分形性
Fractal Tube Formulas for Compact Sets and Relative Fractal Drums: Oscillations, Complex Dimensions and Fractality
AI总结：
本文为任意维欧几里得空间中的相对分形鼓建立逐点与分布分形管公式，用分形 ζ 函数的复维数留数表达邻域体积，并提出基于非实复维数或自然边界的新分形性定义。
摘要：
我们在任意维数的欧几里得空间中，为一大类相对分形鼓建立了逐点和分布意义下的分形管公式。相对分形鼓（简称 RFD）是欧几里得空间中子集构成的有序对 $(A,Ω)$（满足某些温和假设），它推广了（紧）子集以及分形弦的概念。所谓 RFD $(A,Ω)$ 的分形管公式，是指将 $A$ 的 $t$-邻域与 $Ω$ 相交所得体积，表示为某个适当亚纯函数（此处为分形 ζ 函数）在 RFD $(A,Ω)$ 的复维数处留数之和的显式表达式。RFD 的复维数被定义为其经亚纯延拓的分形 ζ 函数（即距离 ζ 函数或管 ζ 函数）的极点；该函数推广了熟知的分形弦几何 ζ 函数。这些分形管公式在很大程度上将此前由第一作者与 van Frankenhuijsen 针对分形弦、随后由第一作者、Pearse 和 Winter 针对分形喷雾所得的相应公式推广到更高维。文中通过若干有趣的例子加以说明，包括分形弦、Sierpiński 垫与三维地毯、分形巢和几何啁啾，以及自相似分形喷雾。我们还提出了一种新的分形性定义：若一个有界集（或 RFD）至少具有一个非实复维数，或其分形 ζ 函数具有自然边界，则称其为分形的。该定义把先前在分形弦背景下引入的定义扩展到 RFD 和 $\u0002mathbb{R}^N$ 中任意有界子集，并通过 Cantor 图（或称魔鬼阶梯）RFD 加以说明；该 RFD 被证明是“次临界分形的”。
英文摘要：
We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,Ω)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,Ω)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $Ω$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,Ω)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpiński gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $\mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devil's staircase) RFD, which is shown to be `subcritically fractal'.
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